Abstract
The solution of higher-dimensional problems, such as the simulation of plasma turbulence in a fusion device as described by the five-dimensional gyrokinetic equations, is a grand challenge for current and future high-performance computing. The sparse grid combination technique is a promising approach to the solution of these problems on large-scale distributed memory systems. The combination technique numerically decomposes a single large problem into multiple moderately-sized partial problems that can be computed in parallel, independently and asynchronously of each other. The ability to efficiently combine the individual partial solutions to a common sparse grid solution is a key to the overall performance of such large-scale computations. In this work, we present new algorithms for the recombination of distributed component grids and demonstrate their scalability to 180, 225 cores on the supercomputer Hazel Hen.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
SG + + library, http://sgpp.sparsegrids.org/
Ali, M.M., Strazdins, P.E., Harding, B., Hegland, M., Larson, J.W.: A fault-tolerant gyrokinetic plasma application using the sparse grid combination technique. In: International Conference on High Performance Computing & Simulation (HPCS), Amsterdam, pp. 499–507. IEEE (2015)
Brizard, A., Hahm, T.: Foundations of nonlinear gyrokinetic theory. Rev. Mod. Phys. 79, 421–468 (2007)
Bungartz, H.J., Griebel, M.: Sparse grids. Acta Numer. 13, 147–269 (2004)
Cappello, F., Geist, A., Gropp, W., Kale, S., Kramer, B., Snir, M.: Toward exascale resilience: 2014 update. Supercomput. Front. Innov. 1 (1), 5–28 (2014)
Dannert, T.: Gyrokinetische Simulation von Plasmaturbulenz mit gefangenen Teilchen und elektromagnetischen Effekten. Ph.D. thesis, Technische Universität München (2004)
Dannert, T., Görler, T., Jenko, F., Merz, F.: Jülich blue gene/p extreme scaling workshop 2009. Technical report, Jülich Supercomputing Center (2010)
Doyle, E.J., Kamada, Y., Osborne, T.H., et al.: Chapter 2: plasma confinement and transport. Nucl. Fusion 47 (6), S18 (2007)
Görler, T., Lapillonne, X., Brunner, S., Dannert, T., Jenko, F., Merz, F., Told, D.: The global version of the gyrokinetic turbulence code GENE. J. Comput. Phys. 230 (18), 7053–7071 (2011)
Griebel, M., Huber, W., Rüde, U., Störtkuhl, T.: The combination technique for parallel sparse-grid-preconditioning or -solution of PDEs on workstation networks. In: Parallel Processing: CONPAR 92 VAPP V. LNCS, vol. 634. Springer, Berlin/New York (1992)
Griebel, M., Schneider, M., Zenger, C.: A combination technique for the solution of sparse grid problems. In: de Groen, P., Beauwens, R. (eds.) Iterative Methods in Linear Algebra. IMACS, pp. 263–281. Elsevier/North Holland (1992)
Heene, M., Pflüger, D.: Efficient and scalable distributed-memory hierarchization algorithms for the sparse grid combination technique. In: Parallel Computing: On the Road to Exascale. Advances in Parallel Computing, vol. 27. IOS Press, Amsterdam (2016)
Heene, M., Kowitz, C., Pflüger, D.: Load balancing for massively parallel computations with the sparse grid combination technique. In: Parallel Computing: Accelerating Computational Science and Engineering (CSE). Advances in Parallel Computing, vol. 25, pp. 574–583. IOS Press, Amsterdam (2014)
Hegland, M., Garcke, J., Challis, V.: The combination technique and some generalisations. Linear Algebra Appl. 420 (2–3), 249–275 (2007)
Hegland, M., Harding, B., Kowitz, C., Pflüger, D., Strazdins, P.: Recent developments in the theory and application of the sparse grid combination technique. In: Proceedings of the SPPEXA Symposium 2016, Garching. Lecture Notes in Computational Science and Engineering. Springer (2016)
Hupp, P., Jacob, R., Heene, M., et al.: Global communication schemes for the sparse grid combination technique. Par. Comput.: Accel. Comput. Sci. Eng. 25, pp. 564–573 (2014)
Hupp, P., Heene, M., Jacob, R., Pflüger, D.: Global communication schemes for the numerical solution of high-dimensional {PDEs}. Parallel Comput. 52, 78–105 (2016)
Kowitz, C., Hegland, M.: The sparse grid combination technique for computing eigenvalues in linear gyrokinetics. Procedia Comput. Sci. 18 (0), 449–458 (2013). 2013 International Conference on Computational Science
Parra Hinojosa, A., Kowitz, C., Heene, M., Pflüger, D., Bungartz, H.J.: Towards a fault-tolerant, scalable implementation of GENE. In: Proceedings of ICCE 2014, Nara. Lecture Notes in Computational Science and Engineering. Springer (2015)
Parra Hinojosa, A., Harding, B., Hegland, M., Bungartz, H.J.: Handling silent data corruption with the sparse grid combination technique. In: Proceedings of the SPPEXA Symposium 2016, Garching. Lecture Notes in Computational Science and Engineering. Springer (2016)
Pflüger, D., Bungartz, H.J., Griebel, M., Jenko, F., et al.: EXAHD: an exa-scalable two-level sparse grid approach for higher-dimensional problems in plasma physics and beyond. In: Euro-Par 2014: parallel processing workshops, Porto. Lecture Notes in Computer Science, vol. 8806, pp. 565–576. Springer International Publishing (2014)
Thakur, R., Rabenseifner, R., Gropp, W.: Optimization of collective communication operations in MPICH. Int. J. High Perform. C. 19, 49–66 (2005)
Acknowledgements
This work was supported by the German Research Foundation (DFG) through the Priority Program 1648 “Software for Exascale Computing” (SPPEXA).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Heene, M., Pflüger, D. (2016). Scalable Algorithms for the Solution of Higher-Dimensional PDEs. In: Bungartz, HJ., Neumann, P., Nagel, W. (eds) Software for Exascale Computing - SPPEXA 2013-2015. Lecture Notes in Computational Science and Engineering, vol 113. Springer, Cham. https://doi.org/10.1007/978-3-319-40528-5_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-40528-5_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-40526-1
Online ISBN: 978-3-319-40528-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)